Curvature and transport inequalities for Markov chains in discrete spaces

We study various transport-information inequalities under three di erent notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Émery [4], the exponential curvature-dimension condition of Bauer et al. [6] and the coarse Ricci curvature of Ollivier [38]. We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an L1 transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet-Myers theorem under the curvature-dimension condition CD(κ,∞) of Bakry and Émery [4].